Solving Inequalities Find The Truth Set Of (3/8)(x+2) ≤ (1/4)(3x-1)

Hey guys! Today, we're diving deep into the world of inequalities and tackling a fascinating problem: finding the truth set of the inequality $\frac{3}{8}(x+2) \leq \frac{1}{4}(3 x-1)$. If you've ever felt a little intimidated by inequalities, don't worry! We're going to break it down step-by-step, making sure you understand not just how to solve it, but why each step works. So, grab your thinking caps, and let's get started!

Understanding the Basics of Inequalities

Before we jump into the specifics of our problem, let's quickly review the fundamentals of inequalities. Inequalities are mathematical statements that compare two expressions using symbols like less than (<), greater than (>), less than or equal to ($\leq$), and greater than or equal to ($\geq$). Unlike equations, which have a single solution or a finite set of solutions, inequalities often have a range of solutions. This range is what we call the "truth set," and our goal is to find all the values of 'x' that make the inequality true.

Think of it like this: imagine a balancing scale. In an equation, the scale is perfectly balanced. In an inequality, the scale is tilted to one side, indicating that one side is "heavier" or "lighter" than the other. Our job is to figure out what values of 'x' keep the scale tilted in the correct direction. Inequalities are a powerful tool in mathematics and are used extensively in various fields, including economics, physics, and computer science. They allow us to model real-world situations where exact equality is not always possible or necessary. For instance, we might use an inequality to describe a budget constraint (spending less than or equal to a certain amount) or a speed limit (driving at or below a certain speed). Understanding the nuances of inequalities is crucial for problem-solving in these diverse contexts.

When working with inequalities, there are a few key rules to keep in mind. Most operations we perform on equations also apply to inequalities, such as adding or subtracting the same value from both sides. However, there's one crucial difference: multiplying or dividing both sides by a negative number reverses the direction of the inequality sign. This is because multiplying or dividing by a negative number flips the order of the number line. For example, if 2 < 3, then multiplying both sides by -1 gives -2 > -3. This seemingly small detail is often a source of errors, so it's something to pay close attention to. Another important concept is the graphical representation of inequalities. We can visualize the solution set of an inequality on a number line. For example, the inequality x > 2 represents all numbers to the right of 2 (excluding 2 itself), while x \leq 5 represents all numbers to the left of 5, including 5. This visual representation can be incredibly helpful in understanding the range of solutions and identifying potential errors in our calculations. In addition to single inequalities, we often encounter compound inequalities, which combine two or more inequalities using the words "and" or "or." For example, "x > 2 and x < 5" represents all numbers that are both greater than 2 and less than 5, while "x < 2 or x > 5" represents all numbers that are either less than 2 or greater than 5. Solving compound inequalities involves solving each individual inequality and then considering the intersection or union of their solution sets, depending on whether the inequalities are joined by "and" or "or."

Step-by-Step Solution: Finding the Truth Set

Now, let's tackle our main problem: $\frac{3}{8}(x+2) \leq \frac{1}{4}(3 x-1)$. Our goal is to isolate 'x' on one side of the inequality. Here's how we'll do it:

1. Eliminate the Fractions

The first step to simplifying this inequality is to get rid of those pesky fractions. To do this, we'll find the least common multiple (LCM) of the denominators, which are 8 and 4. The LCM of 8 and 4 is 8. So, we'll multiply both sides of the inequality by 8:

$8 * \frac{3}{8}(x+2) \leq 8 * \frac{1}{4}(3 x-1)$

This simplifies to:

$3(x+2) \leq 2(3x-1)$

Multiplying both sides of an inequality by the least common multiple is a crucial step in simplifying and solving inequalities that involve fractions. This process eliminates the fractions, making the inequality easier to work with and reducing the likelihood of errors in subsequent steps. The least common multiple (LCM) is the smallest positive integer that is divisible by all the denominators in the inequality. In our case, the denominators are 8 and 4, and their LCM is 8. By multiplying both sides of the inequality by the LCM, we effectively clear the fractions. This is because each fraction's denominator will divide evenly into the LCM, resulting in whole number coefficients. For instance, when we multiply $\frac{3}{8}(x+2)$ by 8, the 8 in the denominator cancels out with the 8 we're multiplying by, leaving us with 3(x+2). Similarly, when we multiply $\frac{1}{4}(3x-1)$ by 8, the 4 in the denominator divides into the 8, resulting in 2(3x-1). This transformation is valid because multiplying both sides of an inequality by a positive number does not change the direction of the inequality sign. However, it's essential to remember that if we were to multiply by a negative number, we would need to reverse the inequality sign. Eliminating fractions not only simplifies the algebraic manipulation but also makes it easier to visualize the relationship between the expressions on both sides of the inequality. It allows us to focus on the core variables and coefficients without the added complexity of fractional terms. This step is particularly beneficial when dealing with more complex inequalities involving multiple fractions or variables.

2. Distribute

Next, we'll distribute the numbers outside the parentheses:

$3x + 6 \leq 6x - 2$

Distributing the constants outside the parentheses is a fundamental step in solving inequalities (and equations) that ensures each term inside the parentheses is properly accounted for. The distributive property states that a(b + c) = ab + ac. In our case, we're applying this property to both sides of the inequality. On the left side, we distribute the 3 across (x + 2), which means we multiply 3 by both x and 2. This gives us 3 * x = 3x and 3 * 2 = 6, resulting in the expression 3x + 6. Similarly, on the right side, we distribute the 2 across (3x - 1), multiplying 2 by both 3x and -1. This yields 2 * 3x = 6x and 2 * -1 = -2, leading to the expression 6x - 2. Distributing correctly is crucial because it removes the parentheses, allowing us to combine like terms and isolate the variable 'x'. If we were to skip this step or perform it incorrectly, we would likely arrive at an incorrect solution. The distributive property is a cornerstone of algebra and is used extensively in simplifying expressions, solving equations, and manipulating inequalities. It is important to understand the underlying principle and apply it carefully to avoid errors. When distributing, it's also essential to pay attention to the signs. For example, if we were distributing a negative number, we would need to be mindful of how it affects the signs of the terms inside the parentheses. A common mistake is to forget to distribute the constant to all terms within the parentheses, especially when there are multiple terms. To avoid this, it can be helpful to write out each multiplication step explicitly, ensuring that every term is accounted for. After distributing, the inequality becomes a simpler form that is easier to manipulate. We have eliminated the parentheses and now have a linear inequality with terms involving 'x' and constant terms. This sets the stage for the next steps in the solution process, which involve isolating 'x' on one side of the inequality.

3. Collect Like Terms

Now, let's get all the 'x' terms on one side and the constants on the other. We'll subtract 3x from both sides:

$3x + 6 - 3x \leq 6x - 2 - 3x$

This simplifies to:

$6 \leq 3x - 2$

Then, we'll add 2 to both sides:

$6 + 2 \leq 3x - 2 + 2$

This simplifies to:

$8 \leq 3x$

Collecting like terms is a pivotal step in solving inequalities, as it consolidates similar terms and simplifies the inequality, bringing us closer to isolating the variable. This process involves rearranging the terms in the inequality so that all terms containing the variable (in this case, 'x') are on one side of the inequality, and all constant terms are on the other side. The goal is to create a more manageable expression that allows us to easily determine the range of values for 'x' that satisfy the inequality. To collect like terms, we use the properties of inequality, which allow us to add or subtract the same value from both sides of the inequality without changing its direction. In our case, we start with the inequality 3x + 6 \leq 6x - 2. To get all the 'x' terms on one side, we subtract 3x from both sides. This eliminates the '3x' term on the left side and combines it with the '6x' term on the right side. The result is 6 \leq 3x - 2. Next, we want to isolate the 'x' term further, so we move the constant term (-2) to the left side. We achieve this by adding 2 to both sides of the inequality. This eliminates the '-2' term on the right side and adds it to the constant term on the left side. This gives us 8 \leq 3x. Collecting like terms not only simplifies the inequality but also makes it easier to visualize the relationship between the variable and the constants. By grouping similar terms together, we reduce the complexity of the expression and make it more straightforward to isolate the variable. This step is essential for solving a wide range of inequalities and equations, and mastering this technique is crucial for success in algebra. When collecting like terms, it's important to pay close attention to the signs of the terms. A common mistake is to incorrectly combine terms with different signs. To avoid this, it can be helpful to rewrite the inequality with the terms rearranged in a clear and organized manner before performing any operations. After collecting like terms, we have a simplified inequality that is much easier to solve. In our case, the inequality 8 \leq 3x is a significant step closer to isolating 'x' and finding the truth set. The next step will involve dividing both sides by the coefficient of 'x' to finally solve for the variable.

4. Isolate x

Finally, to isolate 'x', we'll divide both sides by 3:

$\frac{8}{3} \leq \frac{3x}{3}$

This gives us:

$\frac{8}{3} \leq x$

This can also be written as:

$x \geq \frac{8}{3}$

Isolating 'x' is the final and crucial step in solving an inequality, as it reveals the range of values that satisfy the original inequality. This step involves manipulating the inequality to get 'x' by itself on one side, with a constant value or expression on the other side. To isolate 'x', we use the properties of inequality, which allow us to perform operations on both sides of the inequality while maintaining its validity. In our case, we have the inequality 8 \leq 3x. To isolate 'x', we need to undo the multiplication by 3. This is achieved by dividing both sides of the inequality by 3. Dividing both sides by a positive number does not change the direction of the inequality sign, so we can proceed with confidence. When we divide 8 by 3, we get $\frac{8}{3}$. When we divide 3x by 3, the 3s cancel out, leaving us with x. This results in the inequality $\frac{8}{3} \leq x$. This inequality states that '$\frac{8}{3}$' is less than or equal to 'x', which is the same as saying 'x' is greater than or equal to '$\frac{8}{3}$'. We can also rewrite the inequality as $x \geq \frac{8}{3}$ to make it clearer that we are looking for values of 'x' that are greater than or equal to $\frac{8}{3}$. Isolating 'x' provides us with the solution set for the inequality. It tells us the specific range of values that 'x' can take to make the inequality true. This is the ultimate goal of solving an inequality, and it allows us to understand the constraints or conditions that 'x' must satisfy. When isolating 'x', it's essential to remember the rule about multiplying or dividing by a negative number. If we were to divide or multiply by a negative number, we would need to reverse the direction of the inequality sign. However, in our case, we are dividing by a positive number, so the inequality sign remains the same. After isolating 'x', we have a clear and concise solution that defines the truth set of the inequality. This solution can be represented graphically on a number line, where we would shade all values to the right of $\frac{8}{3}$, including $\frac{8}{3}$ itself (since the inequality includes "equal to"). The solution can also be expressed in interval notation, which is a compact way of representing a set of numbers. In this case, the interval notation for the solution would be $[\frac{8}{3}, \infty)$.

5. Express the Truth Set

The truth set is all values of x that are greater than or equal to $\frac{8}{3}$. In interval notation, this is: $[\frac{8}{3}, \infty)$.

Visualizing the Solution

It's often helpful to visualize the solution on a number line. We'll draw a number line and mark $\frac{8}{3}$ on it. Since our solution includes $x = \frac{8}{3}$, we'll use a closed circle (or square bracket) at $\frac{8}{3}$. Then, we'll shade the region to the right of $\frac{8}{3}$, indicating that all values in that region are part of the truth set. Visualizing the solution on a number line can provide a clearer understanding of the range of values that satisfy the inequality. It helps to solidify the concept of the truth set as a continuous range of numbers rather than just a single value. The number line representation is particularly useful when dealing with more complex inequalities, such as compound inequalities, where there may be multiple intervals or regions that satisfy the conditions.

Real-World Applications of Inequalities

Inequalities aren't just abstract mathematical concepts; they have numerous real-world applications. For instance, they're used in:

• Budgeting: You might use an inequality to determine how much you can spend on groceries each week while staying within your budget.
• Speed Limits: The speed limit on a road is an inequality – you must drive at or below a certain speed.
• Manufacturing: Companies use inequalities to ensure their products meet certain quality standards or tolerances.
• Optimization Problems: Many real-world problems involve finding the maximum or minimum value of a certain quantity, which often requires the use of inequalities.

Understanding inequalities is therefore essential not just for math class, but for navigating many aspects of daily life. They provide a powerful tool for modeling and solving problems involving constraints, limitations, and ranges of values. In the context of budgeting, inequalities can help you determine the maximum amount you can spend on various categories while staying within your overall budget. For example, if you have a monthly budget of $2000 and you want to allocate a maximum of 30% to rent, you can use the inequality rent \leq 0.3 * 2000 to find the maximum rent you can afford. Similarly, in manufacturing, inequalities are used to define acceptable ranges for product dimensions or material properties. For instance, if a metal rod needs to be between 10 and 10.5 centimeters long, this can be expressed as the compound inequality 10 \leq length \leq 10.5. In optimization problems, inequalities are often used to define constraints or limitations on the variables involved. For example, in a linear programming problem, you might have constraints on the amount of resources available or the demand for certain products. These constraints are typically expressed as inequalities, and the goal is to find the values of the variables that maximize or minimize a certain objective function while satisfying all the constraints. The ubiquity of inequalities in real-world applications underscores the importance of mastering the concepts and techniques involved in solving them.

Conclusion

So there you have it! We've successfully navigated the inequality $\frac{3}{8}(x+2) \leq \frac{1}{4}(3 x-1)$ and found its truth set: $x \geq \frac{8}{3}$, or in interval notation, $[\frac{8}{3}, \infty)$. Remember, the key to solving inequalities is to follow the same rules as equations, with the crucial exception of flipping the inequality sign when multiplying or dividing by a negative number. Keep practicing, and you'll become an inequality-solving pro in no time! Guys, I hope this was helpful, and remember, math can be fun! Understanding inequalities is a crucial skill that extends far beyond the classroom. From making informed financial decisions to solving complex engineering problems, the ability to work with inequalities is invaluable. The systematic approach we've outlined – eliminating fractions, distributing, collecting like terms, isolating the variable, and expressing the solution – provides a robust framework for tackling a wide range of inequality problems. By practicing these steps and paying close attention to the rules and nuances of inequalities, you can build confidence and proficiency in this essential area of mathematics. Remember that visual aids, such as number lines, can be incredibly helpful in understanding the solution set and identifying potential errors. And don't be afraid to break down complex problems into smaller, more manageable steps. With patience and persistence, you can master the art of solving inequalities and unlock their power to solve real-world problems.